Resource allocation games: a compromise stable extension of bankruptcy games.

*(English)*Zbl 1280.91013This paper presents an extension of the traditional bankruptcy problem \((N,E,d)\) to the resource allocation problem \((N,E,d, \alpha)\), where \(N\) represents a finite set of claimants, \(E\geq 0\) is the estate which has to be divided among the claimants, \(d=(d_i)_{i\in N}\) is a vector in \(\mathbb R^N_{++}\) with \(d_i\) representing agent \(i\)’s claim on the estate (\(\sum_{i\in N}d_i\geq E\)), and \(\alpha=(\alpha_i)_{i\in N}\) is a vector in \(\mathbb R^N_{++}\) with \(\alpha_i\) describing agent \(i\)’s reward function \(r_i(x_i)=\alpha_ix_i\) (here \(x_i\geq 0\) is a feasible assignment part possible to be given to agent \(i\) restricted by: \(0\leq x_i\leq d_i\) and \(\sum_{i\in N}x_i=E\)). The problem is how to find the “best” feasible assignment vector \(x=(x_i)_{i\in N}\) among the optimal assignments, that is satisfying \(\sum_{i\in N}\alpha_ix_i =\max\). To solve this problem, the authors construct a TU-game \(v^R\) (called a resource allocation game) corresponding to the resource allocation problem \((N,E,d, \alpha)\). It is shown for the game \(v^R\) that it is compromise stable and totally balanced. Furthermore, an explicit expression for the nucleolus of the game \(v^R\) has been derived.

Reviewer: Tadeusz Radzik (Jelenia Góra)

##### MSC:

91A12 | Cooperative games |

91A06 | \(n\)-person games, \(n>2\) |

91B32 | Resource and cost allocation (including fair division, apportionment, etc.) |

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\textit{S. Grundel} et al., Math. Methods Oper. Res. 78, No. 2, 149--169 (2013; Zbl 1280.91013)

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